Electromagnetic Field Theory Study Notes

Course Overview and Administrative Structure

  • Course Title: Electromagnetic Field Theory (EMFT)

  • Instructor: Dr. Abu Bakr Siddique

  • Primary Course Textbook: Field & Wave Electromagnetics by David K. Cheng

  • Secondary Reference Textbook: Electromagnetic Field Theory by Hozorgu and Guru

  • Grading Assessment Scheme:

    • Final Examination: 50%50\%

    • Midterm Examination: 22%22\%

    • Sessional Assessment: 28%28\%

    • Attendance: 8%8\%

    • Homework Assignments: 10%10\%

    • Quizzes: 10%10\%

Curriculum and Topic Breakdown

  • Fundamentals of Electromagnetics & Vector Analysis:

    • Vector Analysis principles and coordinate systems

    • Electrostatics & Magnetostatics foundations: Coulomb's Law, Gauss's Law, Biot-Savart Law, Ampere's Circuit Law, Faraday's Law, and Lenz's Law

  • Maxwell's Equations & Mathematical Modeling:

    • Integral and differential formulations of Maxwell's equations

    • Wave nature of electric and magnetic fields

    • Solutions to Maxwell's equations and constitutive relations

  • Electromagnetic Wave Propagation:

    • Visualization of EM fields and general wave equation solutions

    • Fundamental wave parameters, Poynting vector, and power flow

    • Wave polarization (linear, circular, elliptical)

    • Complex wave parameters and wave behavior in lossy dielectrics, lossless dielectrics, and good conductors

    • Skin depth mechanisms

    • Plane wave behavior at normal and oblique interfaces

    • Formation of standing waves

  • Transmission Line Theory:

    • Lumped-Element circuit model representation

    • Field analysis of transmission lines

    • Terminated lossless transmission lines

    • Quarter-wave transformers and single/double stub matching

  • Antenna Theory & Parameters:

    • Radiation patterns, radiation power density, and radiation intensity

    • Beam width, directivity, antenna efficiency, and gain

    • Beam efficiency, input impedance, polarization, maximum directivity, and maximum effective area

    • Numerical computational techniques for field analysis

  • Antenna Classifications & Measurements:

    • Element types: Hertzian dipole, half-wave dipole, quarter-wave monopole, small loop antennas, and antenna arrays

    • Measurement protocols: gain, impedance, and radiation current measurements

Historical Milestones in Electromagnetic Theory

  • Maxwell's Theoretical Foundation (1865):

    • James Clerk Maxwell mathematically demonstrated that electric and magnetic fields are coupled, propagating through space as waves without needing a physical wire or mechanical medium.

    • Maxwell derived the propagation velocity of these waves and proved it was equal to the speed of light (c≈3×108 m/sc \approx 3 \times 10^8\,\text{m/s}) determined via optical experiments.

    • This unified electricity, magnetism, and optics into a single foundational theory.

  • Hertz's Experimental Validation (1887):

    • Heinrich Hertz experimentally validated Maxwell's wave theory 20 years later by transmitting and detecting wire-free EM waves.

    • Transmitter Mechanism: High-voltage pulses were applied across two metal wires separated by a spark gap, creating high-frequency electric sparks.

    • Receiver Mechanism: Sparks radiated high-frequency EM waves through air, inducing an electric current and a secondary spark across the gap of a loop receiver positioned over a meter away.


    Hertz Spark Gap Setup

  • Marconi's Wireless Communication Systems:

    • Guglielmo Marconi extended Hertz's lab demonstration by engineering transmitter and antenna systems capable of radiating EM waves over distances spanning many kilometers.

    • Marconi's work inaugurated long-distance radio communication.

Electromagnetic Wave Dynamics and Structure

  • Mechanism of Generation:

    • Oscillating or accelerating electrical charges create time-varying electric fields (E\mathbf{E}) and magnetic fields (B\mathbf{B}) that radiate outward.

    • The frequency of the emitted EM wave equals the angular oscillation frequency of the charge source.

  • Field Orientation and Wave Characteristics:

    • EM waves are Transverse Electromagnetic (TEM) waves in free space.

    • Electric field E\mathbf{E} and magnetic field B\mathbf{B} oscillate perpendicular to each other and perpendicular to the direction of propagation.


    EM Wave Vector Diagram

Electromagnetic Spectrum and Frequency Classifications

  • Low-Frequency Radiation:

    • Features longer wavelengths and lower photon energies.

    • Encompasses long radio waves (television and audio broadcasts), microwaves (telecommunications and dielectric heating), and infrared radiation (thermal emissions).


    Infrared Heat Waves

  • Visible Light Region:

    • Occupies a narrow band near the center of the spectrum, emitted by stellar bodies, chemical reactions, and electronic displays.

  • High-Frequency Radiation:

    • Features short wavelengths, high frequencies, and high photon energies.

    • Encompasses Ultraviolet (UV), X-rays, and Gamma rays.

    • Ionizing high-frequency waves pass through human biological tissues, enabling medical imaging, airport security scans, and radiation therapies.


    High Frequency Applications

Radio Frequency (RF) Spectrum Bands

  • Spectrum Specifications and Applications:

    • Very Low Frequency (VLF):

    • Frequency range: 3 kHz−30 kHz3\,\text{kHz} - 30\,\text{kHz}

    • Wavelength range: 100 km−10 km100\,\text{km} - 10\,\text{km}

    • Applications: Submarine communications, near-field communication (NFC/RFID), low-frequency audio broadcasting.

    • Low Frequency (LF):

    • Frequency range: 30 kHz−300 kHz30\,\text{kHz} - 300\,\text{kHz}

    • Wavelength range: 10 km−1 km10\,\text{km} - 1\,\text{km}

    • Applications: Maritime radio navigation and beacon transceivers.

    • Medium Frequency (MF):

    • Frequency range: 300 kHz−3 MHz300\,\text{kHz} - 3\,\text{MHz}

    • Wavelength range: 1 km−100 m1\,\text{km} - 100\,\text{m}

    • Applications: AM radio broadcasts, aviation communication, maritime radio.

    • High Frequency (HF):

    • Frequency range: 3 MHz−30 MHz3\,\text{MHz} - 30\,\text{MHz}

    • Wavelength range: 100 m−10 m100\,\text{m} - 10\,\text{m}

    • Applications: Shortwave radio broadcasts, weather stations, long-range skywave communications.

    • Very High Frequency (VHF):

    • Frequency range: 30 MHz−300 MHz30\,\text{MHz} - 300\,\text{MHz}

    • Wavelength range: 10 m−1 m10\,\text{m} - 1\,\text{m}

    • Applications: FM radio broadcasts, legacy analog television, VHF air traffic control.

    • Ultra High Frequency (UHF):

    • Frequency range: 300 MHz−3 GHz300\,\text{MHz} - 3\,\text{GHz}

    • Wavelength range: 1 m−100 mm1\,\text{m} - 100\,\text{mm}

    • Applications: Modern cellular communication (GSM, CDMA, LTE), Wi-Fi, Bluetooth, GPS networks, pagers, digital television.

    • Super High Frequency (SHF):

    • Frequency range: 3 GHz−30 GHz3\,\text{GHz} - 30\,\text{GHz}

    • Wavelength range: 100 mm−10 mm100\,\text{mm} - 10\,\text{mm}

    • Applications: Point-to-point microwave communication links, satellite transmission, 5 GHz5\,\text{GHz} Wi-Fi, radar, microwave ovens.

    • Extremely High Frequency (EHF):

    • Frequency range: 30 GHz−300 GHz30\,\text{GHz} - 300\,\text{GHz}

    • Wavelength range: 10 mm−1 mm10\,\text{mm} - 1\,\text{mm}

    • Applications: Radio astronomy, satellite meteorology, high-speed 5G5\text{G} wireless backhaul networks.


    RF Frequency Spectrum Breakdown

  • Bandwidth vs. Attenuation Trade-offs:

    • Information-carrying capacity (bandwidth) increases at higher operating frequencies.

    • Higher frequencies experience greater atmospheric attenuation and signal degradation, requiring strict line-of-sight (LoS) alignment.

Transceiver Systems and Wireless Links

  • Transmitting Chain:

    • An alternating current (AC) signal generated by a transmitter moves along a guided transmission line to a transmitting antenna (TxTx).

    • The transmitting antenna accelerates charges, transforming the electrical current into a radiating EM wave.

  • Free Space Propagation & Receiving Chain:

    • The radiated wave travels through space as a free electromagnetic wave.

    • Upon striking the receiving antenna (RxRx), the wave induces oscillating currents, converting the wave back into an AC electrical signal inside the receiver circuit.


    Radio Transceiver System Link

Vector Analysis and Algebraic Properties

  • Scalar vs. Vector Quantities:

    • Scalar: Defined strictly by magnitude (e.g., time, temperature, distance, mass).

    • Vector: Defined by both magnitude and spatial direction (e.g., force, acceleration, velocity, displacement).

    • Notation: Bold symbol A\mathbf{A} or overhead arrow A⃗\vec{A}. Magnitude is written as ∣A∣|\mathbf{A}| or AA.

  • Vector Inversion:

    • −A-\mathbf{A} is a vector having the same magnitude as A\mathbf{A} but pointing in the opposite direction.


    Negative Vector Diagram

  • Vector Addition Laws:

    • Head-to-Tail Rule: Align the tail of B\mathbf{B} with the head of A\mathbf{A}. The resultant sum vector A+B\mathbf{A} + \mathbf{B} originates at the tail of A\mathbf{A} and ends at the head of B\mathbf{B}.

    • Commutative Law:     A+B=B+A\mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A}

    • Associative Law:     (A+B)+C=A+(B+C)(\mathbf{A} + \mathbf{B}) + \mathbf{C} = \mathbf{A} + (\mathbf{B} + \mathbf{C})

  • Vector Subtraction:

    • Performed by adding the negated vector:     A−B=A+(−B)\mathbf{A} - \mathbf{B} = \mathbf{A} + (-\mathbf{B})

  • Multiplication by a Scalar:

    • Multiplying vector A\mathbf{A} by a positive scalar aa scales its magnitude by aa while keeping direction unchanged.

    • If a<0a < 0, vector magnitude scales by ∣a∣|a| and direction flips.

    • Distributive Property:     a(A+B)=aA+aBa(\mathbf{A} + \mathbf{B}) = a\mathbf{A} + a\mathbf{B}

Vector Products: Dot and Cross Products

  • Dot Product (Scalar Product):

    • Definition:     A⋅B≡ABcos⁡(θ)\mathbf{A} \cdot \mathbf{B} \equiv A B \cos(\theta)     where θ\theta is the angle between A\mathbf{A} and B\mathbf{B} placed tail-to-tail (0≤θ≤π0 \le \theta \le \pi).

    • The dot product yields a scalar value.

    • Commutative Property:     A⋅B=B⋅A\mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A}

    • Distributive Property:     A⋅(B+C)=A⋅B+A⋅C\mathbf{A} \cdot (\mathbf{B} + \mathbf{C}) = \mathbf{A} \cdot \mathbf{B} + \mathbf{A} \cdot \mathbf{C}

    • Geometric Interpretation: A⋅B\mathbf{A} \cdot \mathbf{B} equals magnitude AA multiplied by the orthogonal projection of B\mathbf{B} along A\mathbf{A} (or magnitude BB times projection of A\mathbf{A} along B\mathbf{B}).

    • Special Cases:

    • Parallel vectors (θ=0\theta = 0): A⋅B=AB\mathbf{A} \cdot \mathbf{B} = AB

    • Self dot product: A⋅A=A2\mathbf{A} \cdot \mathbf{A} = A^2

    • Perpendicular vectors (θ=90∘\theta = 90^\circ): A⋅B=0\mathbf{A} \cdot \mathbf{B} = 0

  • Cross Product (Vector Product):

    • Definition:     A×B≡ABsin⁡(θ) n^\mathbf{A} \times \mathbf{B} \equiv A B \sin(\theta)\,\hat{\mathbf{n}}     where n^\hat{\mathbf{n}} is a unit vector perpendicular to the plane containing vectors A\mathbf{A} and B\mathbf{B}.

    • Magnitude is given by ABsin⁡(θ)AB \sin(\theta).

    • Direction of n^\hat{\mathbf{n}} is governed by the Right-Hand Rule: Point index finger along A\mathbf{A}, curl fingers toward B\mathbf{B}, and the extended thumb points in direction n^\hat{\mathbf{n}}.

    • Anti-commutative Property:     A×B=−(B×A)\mathbf{A} \times \mathbf{B} = -(\mathbf{B} \times \mathbf{A})

Vector Analysis in Cartesian Components

  • Cartesian Basis Representation:

    • Let x^\hat{\mathbf{x}}, y^\hat{\mathbf{y}}, and z^\hat{\mathbf{z}} be orthogonal unit vectors parallel to the xx, yy, and zz axes.

    • Vector expression:     A=Axx^+Ayy^+Azz^\mathbf{A} = A_x \hat{\mathbf{x}} + A_y \hat{\mathbf{y}} + A_z \hat{\mathbf{z}}

    • Component terms AxA_x, AyA_y, and AzA_z represent the orthogonal projections of A\mathbf{A} along coordinate axes.

  • Component Addition & Scaling:

    • Vector sum:     A+B=(Ax+Bx)x^+(Ay+By)y^+(Az+Bz)z^\mathbf{A} + \mathbf{B} = (A_x + B_x)\hat{\mathbf{x}} + (A_y + B_y)\hat{\mathbf{y}} + (A_z + B_z)\hat{\mathbf{z}}

    • Scalar multiplication:     aA=(aAx)x^+(aAy)y^+(aAz)z^a\mathbf{A} = (a A_x)\hat{\mathbf{x}} + (a A_y)\hat{\mathbf{y}} + (a A_z)\hat{\mathbf{z}}

  • Cartesian Dot Product:

    • Orthonormal basis relations:     x^⋅x^=y^⋅y^=z^⋅z^=1\hat{\mathbf{x}} \cdot \hat{\mathbf{x}} = \hat{\mathbf{y}} \cdot \hat{\mathbf{y}} = \hat{\mathbf{z}} \cdot \hat{\mathbf{z}} = 1     x^⋅y^=x^⋅z^=y^⋅z^=0\hat{\mathbf{x}} \cdot \hat{\mathbf{y}} = \hat{\mathbf{x}} \cdot \hat{\mathbf{z}} = \hat{\mathbf{y}} \cdot \hat{\mathbf{z}} = 0

    • Resulting dot product expansion:     A⋅B=AxBx+AyBy+AzBz\mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z

  • Cartesian Cross Product:

    • Basis unit vector cross products:     x^×x^=y^×y^=z^×z^=0\hat{\mathbf{x}} \times \hat{\mathbf{x}} = \hat{\mathbf{y}} \times \hat{\mathbf{y}} = \hat{\mathbf{z}} \times \hat{\mathbf{z}} = 0     x^×y^=z^,y^×z^=x^,z^×x^=y^\hat{\mathbf{x}} \times \hat{\mathbf{y}} = \hat{\mathbf{z}}, \quad \hat{\mathbf{y}} \times \hat{\mathbf{z}} = \hat{\mathbf{x}}, \quad \hat{\mathbf{z}} \times \hat{\mathbf{x}} = \hat{\mathbf{y}}

    • Matrix Determinant Form:     A×B=(x^y^z^AxAyAzBxByBz)\mathbf{A} \times \mathbf{B} = \begin{pmatrix} \hat{\mathbf{x}} & \hat{\mathbf{y}} & \hat{\mathbf{z}} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{pmatrix}

    • Expanded Cartesian Component Form:     A×B=(AyBz−AzBy)x^+(AzBx−AxBz)y^+(AxBy−AyBx)z^\mathbf{A} \times \mathbf{B} = (A_y B_z - A_z B_y)\hat{\mathbf{x}} + (A_z B_x - A_x B_z)\hat{\mathbf{y}} + (A_x B_y - A_y B_x)\hat{\mathbf{z}}

Right-Triangle Trigonometry

  • Triangle Definitions:

    • A right triangle consists of one right angle (90∘90^\circ) and two acute angles (<90∘<90^\circ).

    • Hypotenuse is the longest side opposite the right angle.

    • Opposite is the side across from reference angle θ\theta.

    • Adjacent is the side adjacent to reference angle θ\theta.


    Right Triangle Geometric Layout

  • Core Ratios:   sin⁡(θ)=OppositeHypotenuse\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}   cos⁡(θ)=AdjacentHypotenuse\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}   tan⁡(θ)=OppositeAdjacent=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\sin(\theta)}{\cos(\theta)}

  • Trigonometric Evaluation Example:

    • Consider a right triangle with adjacent side zz and opposite side rr.

    • Hypotenuse expression via Pythagorean theorem:     Hypotenuse=z2+r2\text{Hypotenuse} = \sqrt{z^2 + r^2}

    • Cosine function:     cos⁡(θ)=AdjacentHypotenuse=zz2+r2\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{z}{\sqrt{z^2 + r^2}}

    • Sine function:     sin⁡(θ)=OppositeHypotenuse=rz2+r2\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{r}{\sqrt{z^2 + r^2}}

    • Tangent function:     tan⁡(θ)=sin⁡(θ)cos⁡(θ)=OppositeAdjacent=rz\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{r}{z}